Holomorphic Principal Bundles Over Elliptic Curves III: Singular Curves and Fibrations

Let G be a simple and simply connected complex linear algebraic group. In this paper, we discuss the generalization of the parabolic construction of holomorphic principal G-bundles over a smooth elliptic curve to the case of a singular curve of arithmetic genus one and to a fibration of Weierstrass cubics over a base B. Except for G of type E_8, the method gives a family of weighted projective spaces associated to a sum of line bundles over B. Working with the universal family of Weierstrass curves over affine two-space and its natural C^*-action, we determine the line bundles that arise in the direct sum in terms of the Casimir weights of the group. We show that in the case of a cuspidal or nodal curve C, provided that G is not of type E_8 or C is not cuspidal, the parabolic construction is closely related to constructions of Kostant and Steinberg of sections for the adjoint quotient morphism of the group G or its Lie algebra, and that moreover it gives a compactification of the adjoint quotient. We describe how the construction may be modified for E_8. Finally, given an irreducible representation of G, we characterize the G-bundles arising from the parabolic construction such that the associated vector bundle is unstable.